Finite-size corrections to the correlation function of the spherical model at d >= 4.

Abstract

Finite-size effects in the correlation function G(R,T;L)\ensuremath{\equiv}〈\ensuremath{\Phi}(0)\ensuremath{\cdot}\ensuremath{\Phi}(R)〉 of a spherical-model ferromagnet, confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\ensuremath{'}}$ (d\ensuremath{\ge}4, ${\mathit{d}}^{\ensuremath{'}}$\ensuremath{\le}2) and subjected to twisted boundary conditions, are analyzed. Focusing attention on the region of first-order phase transition (T${\mathit{T}}_{\mathit{c}}$), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon}(=R/L). Two complementary methodologies are employed: (i) dimensional regularization involving the limit d\ensuremath{\rightarrow}4 as approached from the mean-field regime at short separations \ensuremath{\varepsilon}\ensuremath{\ll}1, and (ii) \ensuremath{\zeta}-function regularization at comparably larger separations \ensuremath{\varepsilon}=O(1) using generators from more simplified classes of sums so as to properly handle the singular features of the correlation function at d=4. Results following from the two methodologies are found to be completely consistent with one another.